Title: A Per-Component Diagnostic Protocol for Neural HJB-PIDE Solvers under Control-Dependent Lévy Jumps

Abstract: This study introduces a five-step diagnostic framework designed for residual-trained neural solvers of Hamilton-Jacobi-Bellman partial integro-differential equations (HJB-PIDEs) featuring control-dependent Lévy jumps. The protocol addresses a prevalent failure mode in neural PDE methodologies, wherein a learned solution may satisfy standard scalar diagnostics yet fail to accurately compute specific operators within its training loss. Our approach necessitates pairing each neural solution with at least one independent, from-scratch reference calculation. It involves decomposing the Hamiltonian into its constituent parts—drift, diffusion, compensator, and nonlocal integral—across a u-grid, and subsequently comparing the value function alongside its low-order derivatives over a (t,x) grid prior to any argmax evaluation.

When applied to the standard CRRA-Merton-Variance-Gamma benchmark, the protocol identified a critical omission: a missing 1/2-mixture factor in the neural method’s importance-proposal density. This error resulted in the nonlocal integral being scaled by exactly one-half, a classic indicator of a constant proposal scale error that remains undetected by extended training, grid refinement, or truncation sweeps. Upon rectifying this bug, four distinct reference methods—comprising two finite-difference solvers with separate discretizations, the corrected neural solver, and a semi-analytic scalar baseline derived from CRRA homogeneity—achieved agreement on the optimal control to within approximately 2%.

While the constant-coefficient CRRA benchmark simplifies to a scalar maximization problem via homogeneity, making the scalar baseline the most efficient verification tool, the primary contribution of this work is the diagnostic protocol itself. This framework is theoretically applicable to more complex, non-homogeneous, and higher-dimensional scenarios where neural HJB-PIDE solvers are essential. This case study highlights a broader verification challenge in neural PDEs: pointwise consistency of a learned value or control can mask systematic errors in the nonlocal operator. Consequently, per-component and surface-level checks are imperative before relying on the argmax policy.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC